2)Department of Aircraft Mechanical Engineering, Inha Technical College, Incheon 402â752, Republic of Korea. 3)Department of Aerospace Engineering, Korea ...
Trans. Japan Soc. Aero. Space Sci. Vol. 58, No. 1, pp. 23–29, 2015
Development of an Aircraft Auto-landing Guidance System using Time Delay Control, and Simulation with Crosswind and Aileron Fault* Hyoung Sik CHOI,1) Sangjong LEE,2) Jangho LEE,1) Hyeok RYU,1) Hyunchul SHIM,3) and Cheolkeun HA4) 1) Korea Aerospace Research Institute, Daejeon 305–806, Republic of Korea Department of Aircraft Mechanical Engineering, Inha Technical College, Incheon 402–752, Republic of Korea 3) Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305–701, Republic of Korea 4) Department of Aerospace Engineering, University of Ulsan, Ulsan 680–749, Republic of Korea 2)
A design procedure for aircraft auto-landing guidance using time delay control (TDC) is proposed, and the designed guidance system is validated via simulations of general landing procedures, including glide slope, flare and touchdown. The simulations adopt a nonlinear dynamics model that includes the landing gear. Additionally, by employing TDC, a stability and control augmentation system, a longitudinal auto-landing guidance law, and a lateral guidance law considering crosswind are proposed. The designed auto-landing guidance laws are evaluated through simulations with aileron fault and crosswind disturbance. The proposed TDC-based auto-landing guidance law shows good performance and is robust when aileron fault and crosswind are included. Key Words:
Aircraft, Auto-landing, Time Delay Control, Aileron Fault, Crosswind
Nomenclature
1.
¡, ¢: angle of attack and sideslip º, ª: roll and pitch angle u, v: x, y velocity components in body axis p, q: roll and pitch rate h: altitude d: lateral offset from reference line V: aircraft velocity in earth frame VN , VE : north and east ground velocity rolling and pitching moment l, M: ¼: heading angle 0 : course angle, tan¹1(VE /VN ) Me : elevator control moment la : aileron control moment Ix;v : moment of inertia in roll, pitch axis $: line of sight angle a , e : aileron and elevator deflection angle £: flight path angle g: gravity acceleration parameter L: small time step t: time e: error vector u: control input vector x: states vector r: reference input Subscripts 0: initial d: desired r: reference command
Time delay control (TDC) is a nonlinear model reference control (MRC) that has recently been applied to various nonlinear plants and models with uncertainty in many engineering fields, including mechanics and robotics, and has shown satisfactory performance.1–5) TDC does not require an accurate model, and is known to be robust against disturbances and for modeling uncertainties.3–5) This technique takes values from the previous step and uses them for feedback. Similar nonlinear control techniques that utilize time delayed observation for feedback have been used6) prior to the late 1980s when Youcef-Toumi and Ito, and Hsia compiled the theories and established this technique. Youcef-Toumi and Ito first named this technique TDC,7,8) while Hsia proposed a similar methodology at around the same time9,10) without using the same nomenclature. Although their works were slightly different in terms of mathematical formulation, both approaches were identical in the concept that they use time delay for estimation. The similarities and differences between their approaches are clearly summarized in Chang and Lee.11) Later, Youcef-Toumi and Wu12) applied stability analysis and input/output feedback linearization (IOFL) control to the TDC using a similar formulation as Hsia and Gao.10) The common aspect of the TDCs from Youcef-Toumi and Ito, and Hsia, and the IOFL TDC from Youcef-Toumi and Wu is that they establish plants in the phase variable (canonical) form. The strength of this approach is that uncertainties and disturbances can be eliminated using very simple formulations. The present paper deals with longitudinal and lateral TDCbased auto-landing guidance laws for an unmanned aerial vehicle (UAV). The authors have previously applied TDC
© 2015 The Japan Society for Aeronautical and Space Sciences + Received 16 April 2012; final revision received 12 March 2013; accepted for publication 23 October 2013. 23
Introduction
Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 1, 2015
^ hðx; tÞ ¼ hðx; t LÞ
for longitudinal auto-landing guidance, and have demonstrated satisfactory robustness of the system against uncertainties in the model and disturbances.13) Further, the feasibility of employing TDC in an actual system was shown through flight tests using TDC SCAS.14) The present paper extends the series of earlier efforts to lateral control loop using the line of sight (LOS) guidance law, and concludes with an auto-landing guidance system. The performance of the TDC-based auto-landing guidance law is evaluated using simulations including aileron fault and steady crosswind. 2.
ð7Þ
Applying Eq. (7), Eq. (3) can be rearranged as: ^ ^ hðx; t LÞ ¼ xðt _ LÞ Buðt hðx; tÞ ¼ LÞ ð8Þ ^ tÞ and Substituting Eq. (8) into Eq. (6) eliminates hðx; yields the inversed control input as: ^ þ ½vðtÞ xðt _ LÞ u ¼ uðt LÞ þ B
ð9Þ
Similar to determining the model inversion control, a feasible pseudo control requires error dynamics, such as Eq. (10). The objective of this control is to make the error converge to zero. Therefore, the desired error dynamics introduces a stable matrix, Ad :
TDC
In this section, the concept of TDC is introduced with a formulation that combines the two previous approaches.8,10) The relationship between two approaches are well summarized in Chang and Lee.11) 2.1. Introduction to TDC Consider a time varying nonlinear dynamic equation with a disturbance, d(t), as11):
^ tÞ = hðx; tÞ, and replacing x_ with v in Assuming hðx; Eq. (10), the pseudo control input becomes:
x_ ¼ fðx; tÞ þ Bðx; tÞ þ dðtÞ
Substituting Eq. (11) into Eq. (9) yields Eq. (12), which is of the same form as shown in Hsia and Gao.10)
e_ d ¼ x_ d x_ ¼ Ad ed
v ¼ x_ d Ad e
ð1Þ
If the exact value of every term in Eq. (1) is known and Bðx; tÞ is perfectly inverse transformable, the input, u, can be found by inverting the model as: ð2Þ u ¼ B1 ðx; tÞ x_ fðx; tÞ dðtÞ
x_ d ¼ Ad xd þ Bd r
ð13Þ
Substituting Eq. (13) into Eq. (12) yields Eq. (14), which is of the same form as shown in Youcef-Toumi and Ito8): _ LÞ u ¼ uðt LÞ þ B^ þ ½Ad x þ Bd r xðt
ð14Þ
2.2. TDC stability overview In this section, a brief overview of TDC stability is presented. Combining Eq. (3) and Eq. (9), the closed loop dynamics can be as Eq. (15)
ð3Þ
^ ^ þ ½vðtÞ xðt _ LÞ x_ ¼ hðx; tÞ þ Buðt LÞ þ B^ B
ð15Þ
^þ
^ B ¼ I, the closed loop Applying Eq. (8) to Eq. (15) and if B dynamics can be defined as: x_ ¼ "ðx; tÞ þ v
With the pseudo control, v, defined in Eq. (5), the input computed by inverting the model yields Eq. (6), which rep^ tÞ resents the typical form of model inversion control. If hðx; = hðx; tÞ, x_ in Eq. (3) is identical to v. However, it is difficult to establish hðx; tÞ exactly. ^ ^ v hðx; tÞ þ Bu ð5Þ ^ tÞ ð6Þ u ¼ B^ þ v hðx;
ð16Þ
where "ðx; tÞ ¼ hðx; tÞ hðx; t LÞ
and v ¼ Ad x þ Bd r:
As presented, the TDC stability takes two main assumptions such that L is sufficiently small and B^ is a proper esti^ þ ¼ I. mation of Bðx; tÞ including the condition, B^ B þ ^ ^ First, the condition, BB ¼ I, Youcef-Toumi and Ito8) proved that the matching condition is always satisfied when the plant system is represented in the phase variable (canonical) form. Second, Youcef-Toumi and Wu12) proved that if L ! 0 and if kI Bðx; tÞB^ 1 k < 1, then the closed loop dynamics is stable. In addition, in Eq. (16), it can be seen that, if L ! 0, then " ! 0, the closed loop dynamics follows the stable desired dynamics Ad , Bd and the error goes to zero
T 1 T ^þ ¼ B ^ : ^ B^ B B
^ tÞ can be defined based on the Here, the unknown hðx; time delay estimation, as shown in Eq. (7). The sampling time step, L, is assumed to be very small and have little difference between the previous and current values for very small magnitudes. ©2015 JSASS
ð12Þ
Furthermore, the desired dynamics can be defined as Eq. (13) applying the same stable Ad of the error dynamics
^ a constant or a matrix with constant elements, is an where, B, estimate of the control effectiveness Bðx; tÞ within a predictable bound. A new nonlinear equation (hðx; tÞ), which includes all the nonlinearities, disturbances and uncertainties, is expressed as: ^ u þ dðtÞ hðx; tÞ ¼ fðx; tÞ þ Bðx; tÞ B ð4Þ
where,
ð11Þ
_ LÞ u ¼ uðt LÞ þ B^ þ ½x_ d Ad ed xðt
However, in reality, direct inverse transformation is not reliable because the exact value of fðx; tÞ and Bðx; tÞ cannot be determined. Therefore, a pseudo control (vðtÞ) is employed and the model is sequentially inverted. First, Eq. (1) is rearranged as: ^ x_ ¼ hðx; tÞ þ Bu
ð10Þ
24
Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 1, 2015
in steady state. However, defining the range of sufficient small L is highly complicated because it is related with hðx; tÞ, which is time variant uncertain nonlinear dynamics. For simplification, Youcef-Toumi and Ito7) suggested that the range of L could be L 0:1=!b in a linear dynamics case. !b is the maximum possible bandwidth of the plant. In general, the UAV flight control computer has fixed update rates due to limited computing power. Therefore, this paper focuses on the practical application of TDC to aircraft and examines the performance of applied TDC with given sample time L(0.025 s). 3.
Fig. 1. Pitch channel SCAS.
where, Ad ¼
Stability/Control Augmentation System (SCAS)
q_ ¼
1 _ . . .Þ Mðq; e ; ; ; Iy
ð17Þ
ð18Þ
Bd ¼
and
0 !2n
# :
p_ ¼
1 lðp; a ; ; . . .Þ Ix
ð24Þ
The TDC control law for the roll channel can be obtained as: 1 € LÞ þ !2 ðr Þ 2!n _ a ¼ a ðt LÞ þ l^ ðt
ð19Þ
n
ð25Þ ^ ^ ^ where, l ¼ la =I x : Table 1 shows the SCAS design parameters for pitch and roll.
where, Mð0 Þ ¼ Mð Þ =Iy : On the other hand, the desired dynamics are defined as: ( ) " #( ) " # 0 1 0 d _d ¼ þ r ð20Þ 2 !n 2!n !2n _d €d
4. where, r is the reference pitch angle input. Also, Eq. (20) is equivalent to Eq. (21):
Auto-landing Guidance
The design of the external loop for auto-landing guidance is discussed in this section. The longitudinal altitude control
ð21Þ
where, ¦ is the damping ratio and !n is the natural frequency of the desired dynamics. Substituting the reference input, r, and the control input, u, in Eq. (14) with their corresponding symbols yields:
©2015 JSASS
2!n
_ ¼ p;
e
_ LÞ e ¼ e ðt LÞ þ B^ þ ½Ad x þ Bd r xðt
!2n
"
^ 1 B^ T : ^ þ ¼ ðB ^ T BÞ ^ e =I^y and B where, m^ ¼ M In Eq. (23), the only property required from the plant ^ which can model is the estimated control effectiveness, m, be obtained by offline calculation. The other terms are all known. Figure 1 shows the pitch SCAS of TDC. Both the damping ratio and the natural frequency of desired dynamics should be determined during tuning. The controller includes a low-pass filter and saturation blocks. The saturation block prevents the control actuator windup.15) The low-pass filter is necessary to eliminate the high-frequency terms induced by the double derivative of ª.3) 3.2. Roll SCAS The roll channel of TDC SCAS can be designed in the same manner as described in the previous section. The pure roll channel dynamics can be defined as:
If the terms including the angle of attack and other terms can be considered as disturbances, Eq. (18) can be expressed as Eq. (19): " # #( ) " # ( ) " 0 1 0 0 _ þ e þ ¼ 0 Mq0 M0 _ t; . . .Þ ð; ; _ €
€d þ 2!n _d þ !2n d ¼ !2n r
1
ð23Þ
is the total pitching moment that is defined as a where M combination of various states and nonlinear tables. Therefore, it shows strong nonlinear characteristics. Equation (17) can be rearranged as a single formulation shown in Eq. (18), with disturbance, ðtÞ: Iy € ¼ Mq q þ Me e þ M þ M_ _ þ ðtÞ þ . . .
#
0
h iT ^ ¼ 0 M _ T and B ^ 0 into Eq. (22) Substituting x ¼ ½ e results in: € LÞ þ !2 ðr Þ 2!n _ e ¼ e ðt LÞ þ m^ 1 ðt n
In this section, the details of TDC-based pitch and roll SCAS design13) are discussed. Since the equations for isolated pitch and roll are in the phase variable (canonical) form, they satisfy the matching condition. Furthermore, these equations are simple. Therefore, they are suitable for the implementation of TDC. 3.1. Pitch SCAS Pure pitch dynamics in the longitudinal motion can be defined as: _ ¼ q;
"
Table 1. SCAS designed parameters. Design parameter ¦ !n
ð22Þ 25
Desired damping ratio (ND) Desired natural frequency (rad/s) Filter cut-off frequency (rad/s)
Pitch
Roll
1.2 3 10
1.2 4 10
Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 1, 2015
employs the TDC-based guidance law,13) while the lateral heading angle control is based on the LOS technique. 4.1. Glide slope guidance The glide slope trajectory is a straight line with a given flight path angle. The altitude for the path can be defined as: h_ ¼ V sin ¼ V
du dt
hd
V
α
{kj zjhz
h
+ θ – α
γ
V sin γ h s
du dt
Delay
KΨ
φr
φ
ωn2 s 2 + 2ζωn s + ωn2
g Vs
Ψ
Fig. 3. Heading controller.16)
ground phase. The present paper assumes that the yaw angle equals the true heading. During the ground run, the nose heading should be controlled using a nose gear steering on the runway. To perform this, the nose gear is mechanically linked with the rudder and the rudder moves proportionally to the roll command. Connecting rudder to roll command is a general control method to reduce adverse yaw due to roll during the flight. The heading controller shown in Fig. 3 assumes that the 2nd-order system inside is the desired roll dynamics of Eq. (19), which is augmented dynamics by the implementation of the TDC SCAS. Using this simplification, the gain, K for the heading control can be easily determined. The root locus generated by the system in Fig. 3 is shown in Fig. 4. This indicates that the system is stable for K less than 1.3. The present paper selects 0.7 for the gain value. With the stable heading control loop being established, the next task is to design heading generation for guidance to the target point. The heading generation employs the LOS technique shown in Fig. 5. The figure depicts the geometry of the lateral guidance that accounts for crosswind. To simplify the problem, the figure shows the situation when the azimuth of the reference line ref equals 0. In the figure, the line of sight is designated by $, the deviation distance from the reference line by d, and the distance to the target by R. The symbols within parentheses denote the definition of the direction. To design a control that makes d exponentially smaller, the desired dynamics of d can be defined as Eq. (33), similar to Eq. (30):
ð28Þ
The block diagram of the glide slope guidance law is presented in Fig. 2. 4.2. Flare guidance The desired flare altitude can be expressed as an exponential function16): ð30Þ
Applying h_ ¼ V sin ¼ V , Eq. (30) can be a solution to Eq. (31): ð31Þ
Finally, the desired flare flight path angle can be determined as: ð32Þ
The criterion for switching from the glide slope phase to the flare phase is determined in terms of the altitude. 4.3. Lateral guidance The lateral guidance is implemented as a heading control. First, the heading controller is designed before dealing with the guidance to the target point. Figure 3 depicts a common heading controller using roll angle control.16) In Figs. 3 and 5, the current heading (¼), which is fed back into this system, requires the ‘course angle’ ( 0 ) during the flight. On the other hand, this system requires the ‘true heading’ (*) during the landing run because the nose heading should be aligned with the runway. Therefore, a feedback heading-switch is required between the flight phase and the ©2015 JSASS
++
1 z
Ψ
ð29Þ
h V
++
・
Ψd
ð27Þ
Appling time delay estimation (TDE) to Eq. (27), ^ LÞ ¼ hðt _ LÞ V^ ðt LÞ and substituting into f^ ¼ fðt Eq. (28), then the TDC glide slope guidance control can be obtained as: 1 1 _ _ ^ glide ¼ glide ðt LÞ þ V hd hðt LÞ þ ðhd hÞ
dflare ¼
τ
Vˆ –1
θr
Fig. 2. Longitudinal glide slope guidance law.
which is exponentially stable, then:
h_ ¼ V ¼ h=
h
++ –
γd
λ s+λ
ð26Þ
1 e_d ¼ ed ;
hdflare ¼ h0flare et=
+–
wGG{kjGzjhz sGˀGG
1 z
where, f^ ¼ V sin V^ þ ðtÞ: If the desired error dynamic is defined as,
1 h_d h_ ¼ h_d V^ f^ ¼ ðhd hÞ
1
h(t – L)
Employing the TDC concept, Eq. (26) can be reformulated as Eq. (27), and £ is considered as an input. h_ ¼ f^ þ V^
・ hd
dd ¼ d0 et=
By applying d_ ¼ V sin
rel
d_ ¼ V
¼V rel
rel ,
ð33Þ Eq. (33) is solved as:
¼ d=
ð34Þ
Finally, the desired heading angle is determined as: reld
¼
d V
ð35Þ
It is notable that Eq. (35) does not have range R in the denominator compare with the geometric LOS guidance law, 1 reld ¼ KLOS tan ðd=RÞ, then Eq. (35) refines the prob26
Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 1, 2015 20 15
Imag Axis
10 5 0 -5 -10 -15 -20 -30
-25
-20
-15
-10 Real Axis
-5
0
5
Fig. 7. EAV-1 AL.
10
Fig. 4. Heading controller root locus (K = 0.7).
Throttle
3 dth(0~1)
XYZ_prop
Thrust
rho
rho
LMN_prop
Vt
Propulsion Vt
States
V
alpha (rad)
Alpha
Ψ
Ψ'
V
3 m/s
d
ψ ref = 0
Λ
ψ = Ψ' ψ =Ψ
ψ rel = ψ
XYZ _prop X
x
Beta
beta (rad)
Crosswind pqr
pqr(rad/s)
rho
L
0.1s+1
2
ψ rel
d
ψ
φ
g Vs
XYZ _LG da (deg )
da(deg)
ψ
States
I.C : x0
0.1s+1 N dr (deg )
V s
LMN_LG
0.1s+1
dr(deg)
ωn2 s 2 + 2ζωn s + ωn2
1 s
LMN_aero
M
1
1
φr
XYZ _aero
de (deg )
1
4
Kψ
Z
1
Ground
Fig. 5. Lateral guidance geometry.
ψd
LMN_prop
x_dot
W ind & Gust rho
Flight de(deg)
ψ ref
Y
Aerodynamics States
d
Equations of Motion including Gravity
F_gr
x
Gear K
Steering
Rudder to nose
1 Vτ
M _gr
Landing gear
Fig. 8. Nonlinear simulation block diagram. 80
Fig. 6. Lateral guidance law. Altitude (m)
60
lem that the loop gain goes to infinity as the range goes to zero while the aircraft approaches near to the target.16) Finally, as shown in Fig. 6, the actual heading guidance command is determined as a sum of the desired heading and the azimuth of the reference line: d
¼
reld
þ
ref
40 20 0
1600
1400
ð36Þ
1200
5.
Modeling and Simulation
1000
North (m)
Figure 7 shows Electric Aerial Vehicle-1 Auto Landing (EAV-1 AL), the test bed of the controller. An aircraft simulation model is developed in the MATLAB/Simulink environment. The refresh rate of the simulation and controller is set to 40 Hz. Therefore, the time step, L, in Eq. (7) becomes 0.025. The aerodynamic data is calculated using DATCOM, and the landing gear model for auto-landing has a tire model with a simple 1st-order spring and damper setup.17) The landing gear model accounts for friction in the longitudinal and lateral directions of the gear wheel so that steering can be simulated. The implemented dynamics model is shown in Fig. 8. 5.1. Crosswind landing simulation Figure 9 shows the 3D trajectory of the TDC-based autolanding guidance simulation. A steady 3 m/s crosswind is applied, starting 2 s after the beginning of the simulation. ©2015 JSASS
800
600
400
200
0
Fig. 9. Crosswind landing flight path (crab, de-crab maneuver).
The feedback heading-switch results in the observable crab and de-crab maneuvers under the crosswind. The flare altitude is set to 1.36 m, which is half the wingspan. The idle setting and the heading-switch for touchdown occur at the flare altitude. 27
East(m), Angle(deg)
Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 1, 2015
Altitude(m)
100 80
Hd
60
H Flare start
40 20
0
20
0 200
400
600
800
1000
1200
1400
1600
1800
3
0
2
5
1 0 1400
1420
1440 1460 Forward ground track (m)
1480
1500
Fig. 10. Longitudinal landing flight path.
1200
1400
1600
1800
0
200
0
200
400 Rudder Steering Flare start
600
800
1000
1200
1400
1600
1800
600
800 1000 North (m)
1200
1400
1600
1800
0
-5
400
0 0
50
100
0
150
50
D(m) 0
0
50
100
150
Elevator(deg)
Az(g) 2 1 0
50
100
V(km/h) 40 20 0
150
3
50
100
150
δ e(deg)
0 -5 -10
150
0
5
0
50
100
150
Time(s)
Time(s)
Fig. 12. Tracking error and touchdown impact.
Fig. 13. Primary longitudinal parameters.
-200
0
-400
East (m)
-200 0
North (m)
-600
-200
-400 -600
-200
East (m)
-800
δ a(deg)
20 10 0 -10
δ e,δ r(deg) θ,φ (deg)
50 0 -50 10 0 -10
50
100
150
200
250
V(km/h)
V
H(m)
0
Hcmd
0
50
100
150
200
H
250
ψ(deg)
200 0 -200
V cmd
ψcmd
0
50
100
150
200
ψ
250
δaL
0
50
100
150
200
δaR
50
100
150
200
250
φ δe
0
50
100
150 Time(s)
200
250
δr
Fig. 16. PID SCAS auto-landing time history. ©2015 JSASS
200 100 0 200 0 -200
250
θ
0
100 50 0
δ a(deg)
H(m)
200 100 0
Fig. 15. TDC SCAS auto-landing trajectory.
δ e,δ r(deg) θ,φ (deg)
V(km/h)
80 60 40
North (m)
-800
Fig. 14. PID SCAS auto-landing trajectory.
ψ (deg)
100
60 Velocity(m)
Lateral offset(m)
1000
50
5
Az(g)
800
H(m)
Altitude(m)
Altitude(m)
0
0
600
100 HError(m)
-5
400 Right gear Left gear Nose gear
Fig. 11. Lateral landing flight path and associated parameters.
5
-5
200
-0.2
Deflection(deg)
Flare (m)
4
0
-0.4
Deflection(cm)
0
Trajectory Heading Roll
-20
20 10 0 -10 20 0 -20 5 0 -5
V cmd
0
50
100
150
200
250
0
50
100
150
200
250
0
50
100
150
200
250
Hcmd H
ψcmd ψ δaL δ
0
50
100
150
200
250 aR
0
50
100
150
200
250
0
50
100
150 Time(s)
200
250
θ φ δe
Fig. 17. TDC SCAS auto-landing time history. 28
V
δr
Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 1, 2015
Figure 10 shows the time histories of the longitudinal auto-landing and flare, and Fig. 11 shows the time histories of the lateral attitudes, landing gear deflection and control angles. Figure 12 is the time histories of the longitudinal and lateral trajectory tracking errors and the impact acceleration during landing. Figure 13 shows the time histories of the altitude, speed and the longitudinal control angle. 5.2. Landing simulation with aileron fault The fault of an aileron is simulated here. For a case where the left aileron is stuck at a 20° angle downward at simulation time 50 s, PID control and TDC control are compared in the auto-landing performance. The simulation consists of a turn, followed by the landing of the aileron-faulted aircraft. Figure 14 shows the landing trajectory of the aircraft with PID-based auto-landing guidance. The PID SCAS is designed using CONDUIT.18) As the fault of the aileron results in rolling moment to the right, the aircraft cannot make a left turn for landing after the first right turn. As a consequence, a safe landing is not achievable. Figure 15 shows the landing trajectory of the aircraft with TDC-based auto-landing guidance. It is remarkable that, for the same fault condition, the aircraft makes a reasonable landing. Figures 16 and 17 show the time histories associated with the PID and TDC controllers, respectively. 6.
time step L is 0.025 s. It confirms that the smaller time step also guaranties satisfactory performance. References 1) Lee, C.-H., Kim, T.-H., and Tahk, M.-J.: Missile Autopilot Design for Agile Turn Using Time Delay Control with Nonlinear Observer, Int. J. Aeronaut. Space Sci., 12 (2011), pp. 266–273. 2) Park, B.-G., Kim, T.-H., and Tahk, M.-J.: Time-Delay Control for Integrated Missile Guidance and Control, Int. J. Aeronaut. Space Sci., 12 (2011), pp. 260–265. 3) Chang, P. H., Park, B. S., and Park, K. C.: An Experimental Study on Improving Hybrid Position Force Control of a Robot Using Time Delay Control, Mechatronics, 6 (1996), pp. 915–931. 4) Chin, S. M., Lee, C. O., and Chang, P. H.: An Experimental Study on the Position Control of an Electrohydraulic Servo System Using Time Delay Control, Control Eng. Practice, 2 (1994), pp. 41–48. 5) Chang, P. H., Kim, D. S., and Park, K. C.: Robust Force/Position Control of a Robot Manipulator Using Time-Delay Control, Control Eng. Practice, 3 (1995), pp. 1255–1264. 6) Morgan, R. G. and Ozguner, U.: A Decentralized Variable Structure Control Algorithm for Robotic Manipulators, IEEE J. Robotics Autom., 1 (1985), pp. 57–65. 7) Youcef-Toumi, K. and Ito, O.: Controller Design for Systems with Unknown Dynamics, American Control Conference, 1987, pp. 836–845. 8) Youcef-Toumi, K. and Ito, O.: A Time Delay Controller Design for Systems with Unknown Dynamics, American Control Conference, 1988, pp. 904–913. 9) Hsia, T. C.: A New Technique for Robust Control of Servo Systems, IEEE Trans. Ind. Electronics, 36 (1989), pp. 1–7. 10) Hsia, T. C. and Gao, L. S.: Robot Manipulator Control Using Decentralized Linear Time-Invariant Time-Delayed Joint Controllers, IEEE International Conference on Robotics and Automation, 1990, pp. 2070–2075. 11) Chang, P. H. and Lee, J. W.: A Model Reference Observer for TimeDelay Control and Its Application to Robot Trajectory Control, IEEE Trans. Control Syst. Technol., 4 (1996), pp. 2–10. 12) Youcef-Toumi, K. and Wu, S.-T.: Input/Output Linearization Using Time Delay Control, ASME J. Dynam. Syst. Meas. Control, 114 (1992), pp. 10–19. 13) Choi, H. S., Lee, S., Lee, J., Kim, E. T., and Shim, H.: Aircraft Longitudinal Auto-Landing Guidance Law Using Time Delay Control Scheme, Trans. Jpn. Soc. Aeronaut. Space Sci., 53 (2010), pp. 207–214. 14) Choi, H. S.: Research on the UAV Model Based Design and Adaptive Control for Uncertain Dynamics, Ph.D. Thesis, KAIST, 2011, pp. 95–97 (in Korean). 15) Chang, P. H. and Park, S. H.: On Improving Time-Delay Control under Certain Hard Nonlinearities, Mechatronics, 13 (2003), pp. 393–412. 16) Blakelock, J. H.: Automatic Control of Aircraft and Missiles, 2nd ed., John Willey & Sons, New York, 1991, pp. 88–97, 178–179. 17) Atkinson, C. J.: Development of an Aerodynamic Table Lookup System and Landing Gear Model for the Cal Poly Flight Simulator, http:// aerosim.calpoly.edu/library/senoir-projects, 2002. 18) Lee, S., Lee, J., and Lee, D. S.: Lateral and Directional SCAS Controller Design Using Multidisciplinary Optimization Program, J. Korean Soc. Aeronaut. Space Sci., 40 (2012), pp. 251–257 (in Korean).
Conclusion
Employing the TDC technique, the present study proposed a design procedure for the auto-landing guidance of an unmanned aerial vehicle (UAV) in both longitudinal and lateral directions. TDC-based controllers for pitch, roll and altitude were designed. The lateral guidance adopted the LOS guidance method with the inside loop of TDC-controlled roll dynamics. The TDC-based auto-landing guidance was validated with the nonlinear simulations for overall landing elements including glide slope, flare and touchdown using a simulation model which included the landing gear model. It was shown that, for the design of control loops which account for the influence of crosswind, crab and de-crab maneuvers can be successfully carried out during the landing procedure by switching the heading feedback. In addition, a comparison simulation for a case with a faulty aileron showed that TDC SCAS is robust and fault tolerant. Additionally, through all the simulations, the applied TDC controller shows satisfactory performance in the case that the
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